Seats and Quotas

LER can incorporate any STV rules for quota and surplus transfers. It enables at least one new hybrid quota.

It is surprising that some people get heated about which quota is “correct”. There are a half dozen in use and, except for the most bizarre, the choice makes little difference. The number of seats effects the accuracy of proportional representation more than the quota rule does. Recall that 5-seat PR districts create fewer “wasted votes” fewer than 3-seat districts. That means fewer votes go to parties (or people) who fail to win a seat, and each party's share of seats is closer to its share of votes. Give the council enough seats so the quota is less than the votes for each significant faction.

(Many organizations find it hard to find enough skilled candidates to fill a 5-seat council – not to mention hard to reward the council members for long hours of study and meetings.)

Table I.

Comparing PR Quotas

Quota

Seats

Rules

3

5

7

9

Formula

Simple
Droop

33.3
25

20
16.7

14.2
12.5

11.1
10

=100/Seats
=100/(Seats+1)

-0.01
+0.01

A fraction of a vote must be added to the Droop quota to make sure the tally does not elect one too many (Seats+1) people. A fraction of a vote may be subtracted from the simple quota so the tally is sure to have enough ballots to elect [seats] people despite any rounding errors.

A ballot with few preferences marked might have all of those candidates eliminated. The ballot may be treated as a tie for all remaining candidates, giving each of them a fraction of its weight; or the ballot and its weight may be ignored and the quota recalculated. A dynamic simple quota is Quota = (Weights Left / Seats Left) -0.01.

This page focuses on optional quotas for electing ensemble councils. Thanks to Dr. T. Nicolaus Tideman, an excellent history of all STV quotes is was available in pdf and text formats.

Plurality rule can leave a large majority of the voters unrepresented, as in the 3-way Korean election of 1987. Majority rules can leave close to half unrepresented, as is common in close, 2-way races.

Thomas Hare developed STV to end such failures by older rules. Hare's simple quota of ballots needed to win a seat made every vote count:
Quota = Voters / Seats.
If there are 100 voters and 5 seats, the quota is 20 and all 100 ballots are needed to elect 5 reps. This is called either the simple quota or the Hare quota. Sometimes the last candidate eliminated is the political opposite of the last candidate elected. Ballots then are forced to transfer to a very low preference or, if that preference is not marked, the ballot cannot be transferred, is thrown out and the last rep is elected with less than a quota. Critics found that simple quota can under represent a majority as shown in case 1 below.

Henry Richmond Droop designed a quota to avoid under representing a majority.
Quota = (Voters / Seats + 1) + 1 vote.
(The term "+ 1 vote" avoids a tie for the last seat.)
This quota has been refined further by Newland and Britton and by Irwin Mann. Most jurisdictions that use STV use one of these three quotas.

Critics point out that, unlike Hare's simple quota, they are designed to leave 1 quota of voters with no rep. The failure of Hare quota to empower the majority can lead to more serious political turmoil than the failure of Droop quota to represent a minority. But Hare's failure may be unlikely and Droop's failure almost certain in a particular electorate.

The easy quota, (Votes+1) / (Seats+1), is easy to remember. It is between Droop and Hare. So it does not fail in quota-borderline cases. (Such cases can make Droop or Hare elect too many or too few winners for the majority. Tideman, Dr. L. Bruce Anderson and others offer many examples to show how this happens. This danger led to the invention of better quota formulas.)

LERa may resolve the debate about which quota is best. It can use Droop quota for the chairperson and can use remaining voters / remaining seats to calculate the Hare quota for other seats. LERa then represents the majority properly while making every vote count. Thus LERa can get the best of both quotas.

Case 1, an election to fill 3 seats.

The candidates were A, B, R, and S and 36 ballots were cast.

Numbers of voters 12 7 9 8
Their preferences AB BA RS SR

Thus 19 voters, a majority, prefer A and B while 17 prefer R and S.

STV

The Hare quota is 36/3=12 votes to win a seat.
With that quota A is elected. No one else has the required quota so the weakest, B, is eliminated. Then R and S are elected. Thus the majority group with 19 voters who prefer A and B get only 1 rep while the 17 who prefer R and S get 2 reps.

The Droop quota is almost 36/4=9 votes.
With more than Droop quota, A is elected and her 3 excess votes transfer to B, who then has 10 and is elected. Then S is eliminated and R is elected. Thus Droop quota protects the majority's right to a majority of the reps.

LERa

Recommended quotas:
Quota for the chair is almost 36/4=9 votes.
Quota for the reps is (36-9)/2=13.5.
If A is the Condorcet winner:
Candidate A gets more than the Droop quota required for the chair and is elected, 3 votes transfer to B. No one gets the Hare quota required for reps so S is eliminated and her ballots transfer. Then R and B are elected.
If B is the Condorcet winner, exempt from elimination:
No one starts with a Hare quota of votes. S is eliminated; her votes transfer to elect R. Then A and B are elected.

The Condorcet winner in Case I (B) had fewer firsts (7) than the last candidate eliminated (S had 8). That causes LERa to switch to LERb if there are more than 3 seats or else to STV. The switch is needed to prevent the problem shown in Case 2. The majority voters gave second rank to the party leader, C, making her the Condorcet winner and chairperson. She is immune from elimination even though she has no firsts. Whether by chance or conspiracy, this shuts out the large minority group.

[The center is nearly always contested by 2 or more candidates. But in many organizations it is not divided or polarized by 2 factions. The voter preferences are more likely to run;

AB BA BR RB RS SR |_____A______B________________R_______S____|

[ Even this is too simple for real life. Most groups do not have one-dimensional politics. The AR, AS, BS, RA, SA, and SB preferences occur. Concocted examples do not include the probabilities and effects of such preferences. Simulations are needed for that next analytical step.]

Case 2 LERa fails, then switches to LERb or STV

The candidates were A, B, C, and R and 100 ballots were cast.

Candidates A B C RFirsts 37 32 0 31

LERa with recommended quotas
The chair's Droop quota is approximately 25 ballots. The simple quota for reps is 37.5 ballots. No one has the required quota so the weakest, R, is eliminated. Then A, B, and C are elected.
STV with recommended quota
The Droop quota is approximately 25 ballots. Candidates A, B, and R are elected.
The Hare quota is approximately 33 ballots. Candidates A, is elected. Four votes transfer to elect B. C is eliminated and R is elected.

Table II: Share of votes needed to survive the last elimination

Seats

Rule

PR Seats

Quota

Min.

Max.

Chair

Notes

3

LERa
LERb
STV Hare
STV Droop

3
2
3
3

37
50
33
25

20
26
17
25

32
34
25
25

25
0
NA
NA

Note 1
Note 2
Note 3
Note 4

5

LERa
LERb
STV Hare
STV Droop

5
4
5
5

21
25
20
17

11
13
11
17

19
21
17
17

17
0
NA
NA

Table II Notes:The quotas above are rounded to the nearest whole number. Droop quotas round down; simple quotas round up.
1) LERa is listed with Droop quota for the chairperson and simple quota for the remaining seats.
Chair quota = Ballots / (Seats+1) + 0.1
Rep quota = (Ballots - Chair quota) / Seats - 0.1
2) LERb has no quota for the chairperson. It is listed with the simple quota for reps. Notice that the maximum needed is about the same as Droop quota for 2 seats.
3) The minimum needed (Min) occurs when all but 1 seat has been filled. Take, for example, a 3 seat election under STV Hare with 2 seats filled. Slightly more than 66% of the ballots were used to elect those 2 winners, so just under 34% of the weight remains. A candidate with more than half of that (17%) will not be eliminated and will win.
The maximum needed (Max) occurs when no one has won and the number of candidates remaining equals the number of seats plus 1. In the case of 3 seats under STV Hare, the last 4 candidates might have 25+, 25, 25, 25-. So 25% is needed to ensure a seat.
4) Notice that the Droop quota is the smallest number of votes that ensures the candidate is within the top [seats] candidates.

Consider the case of a 5 seat council from an electorate with a 35% minority. With almost to two-fifths of the votes they should get 2 of the 5 seats. LERa recommends Droop quota (17%) for the chair and simple quota (21%) for the other seats. They have more than the 21% + 11% minimum needed for 2 seats but less than the maximum 2 × 19%.
Under LERb with simple quota (25% for each of 4 seats) they can win only 1. They would have 10% left over, less than the 13% minimum shown in the table. A 65% majority can win the chair, 2 STV reps, and have 15% left. The majority then wins the last seat and thus 4 of 5 seats.

Endnote: Some jurisdictions might not exemption the Condorcet winner from elimination but give her the advantage of a quota reduced by a gift of weight from all ballots. In that case the candidate with the lowest percentage of her quota is eliminated. If the first CW is eliminated, quota is reduced for the second winner.